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SUMS OF FOUR SQUARES WITH A CERTAIN RESTRICTION

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  14 January 2021

YUE-FENG SHE Open the ORCID record for YUE-FENG SHE [Opens in a new window]  and
HAI-LIANG WU Open the ORCID record for HAI-LIANG WU [Opens in a new window]
YUE-FENG SHE
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China e-mail: she.math@smail.nju.edu.cn
HAI-LIANG WU*
Affiliation:
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China
*
e-mail: whl.math@smail.nju.edu.cn
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Abstract

Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory 175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$. In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.

Keywords

sums of four squaresternary quadratic forms

MSC classification

Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 11E12: Quadratic forms over global rings and fields 11E20: General ternary and quaternary quadratic forms; forms of more than two variables

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 218 - 227
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by the National Natural Science Foundation of China (grant no. 11971222). The second author was supported by NUPTSF (grant no. NY220159).

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